I have a bottle of apple juice that spontaneously created a "mother" (vinegar starter). This is actually the second time that I've had this happen. The first time I had no idea what it was and threw it out. This time I'd like to try making vinegar out of it, but I'm not sure what I need to do.

I have done some google searches but all I can find is making vinegar with a purchased cultures.

Leave it in the bottle of juice and see what happens. You need to leave it in a place friendly to fermentation (so not the fridge), and let it go. Don't cap it tightly, since it will go through alcohol fermentation if it hasn't already - which causes carbonation. Give it the sniff test periodically and see what happens. With a wild mother, you may not get something predictable, but experimenting can be fun.

We make homemade apple cider vinegar (also pear vinegar etc). Just let it sit for a long time and check it every once and a while. It might not ever get as strong as store bought vinegar. I personally just like to drink it bubbly. Mmm!